Why is everyone hung up on this flagship nonsense? The only Enterprise ever called a flagship on screen was the D, and since it wasn't commanded by an admiral, even that was just an honorific. The TOS E was "just one of the fleet," and commanded by the most junior captain in the fleet.
Although I cant off the top of my head without a ton of research demonstrate the original Enterprise was a/the Flagship, I'm almost certain that it was stated to be at some point.
Certainly when Spock was Captain of it during Wrath of Khan, the Enterprise was relegated to training duties and we have absolutely no evidence it was the flagship by that point.
Beyond that, we know Captain Picard was often (relatively speaking) given fleet operation control. Heck, as early as season 1, we know Star Fleet would have gladly given him Admiral posting if he were interested in it. I tend to feel like it was writer's flaw not to more strongly advocate on screen his status as a flag officer. Reading between the lines we're usually led to the assumption that Picard is not a Captain, but rather a Fleet Captain. One might be able to infer that Kirk held the same position.
If we follow the novels, we know it wasn't for some time till John Harriman was officially promoted to Fleet Captain. We never hear if Captain Garrett holds or is promoted to that position ever, in any media.
As far as ship names go; I think anyone and everyone would argue who has ever had any kind of infatuation with a ship that the name and its linage is something very important to most. I suppose one could say star ships are like hurricanes, the really special ones you immortalize; the less then special ones, you recycle the names.
That being said, some of the ship names we've seen wildly jump classes have some real linage behind them. Again, pulling from my initial post, the Defiant has some really strong linage as do ships like the Intrepid, the Excelsior, Voyager (now). Looking back on the Defiant from a linage stand point, it probably would have made sense for the Sovereign Class to be named the Defiant class.